The strong increase in computational power observed during the last few years has allowed to use Large Eddy Simulation (LES) for industrial configurations. Nevertheless, the restitution time is still too large for a daily use in the design phases. The objective of this work is to develop a new time integration method to reduce the restitution time of LES of incompressible flows by allowing the use of larger time-step. The projection method, probably the most commonly used method in the context of LES of incompressible flow, is generally applied using explicit time advancement which constrains the time-step value for stability reasons (CFL and Fourier constraints). The time-step can then be small with respect to the physical characteristic times of the studied flow. In this case, an implicit time advancement method, which is unconditionally stable, can be used. However, this leads to non-linear resolution of momentum equation which can strongly increase restitution time because of non-linear iterations inside a physical iteration. In order to relax the stability constraints while minimizing the computational cost of an iteration, a linearized implicit time advancement based on Backward Differentiation Formula (BDF) scheme is proposed in this work. The linearization is performed using an extrapolated velocity field based on the previous fields. In order to choose a relevant time-step during the calculation, an adaptive time-step method is used. This method consists in evaluating the current local truncation error to estimate the next time-step that aims to provide a user-defined level of error. This new time integration method, which combines implicit linearization and time-step control, is applied to three cases involving a wide range of flows: a round turbulent jet, a turbulent pipe flow, and the PRECCINSTA configuration, a swirled burner which is a representative case of an industrial aeronautical injection system. The new method can lead to a restitution time up to three times lower than the explicit method while keeping the same accuracy in terms of mean and fluctuating velocity fields. Moreover, the new time advancement is incorporated in an automatic mesh convergence procedure and leads to a reduction of the restitution time of around a factor two on the overall procedure.